Breaking hyperbolicity for smooth symplectic toral diffeomorphisms

    2010, Volume 15, Numbers 2-3, pp.  194-209

    Author(s): Lerman L. M.

    We study a smooth symplectic 2-parameter unfolding of an almost hyperbolic diffeomorphism on two-dimensional torus. This diffeomorphism has a fixed point of the type of the degenerate saddle. In the parameter plane there is a bifurcation curve corresponding to the transition from the degenerate saddle into a saddle and parabolic fixed point, separatrices of these latter points form a channel on the torus. We prove that a saddle period-2 point exists for all parameter values close to the co-dimension two point whose separatrices intersect transversely the boundary curves of the channel that implies the existence of a quadratic homoclinic tangency for this period-2 point which occurs along a sequence of parameter values accumulating at the co-dimension 2 point. This leads to the break of stable/unstable foliations existing for almost hyperbolic diffeomorphism. Using the results of Franks [1] on $\pi_1$-diffeomorphisms, we discuss the possibility to get an invariant Cantor set of a positive measure being non-uniformly hyperbolic.
    Keywords: symplectic diffeomorphism, torus, Anosov diffeomorphism, almost hyperbolic, degenerate saddle, bifurcation, homoclinic tangency, break of hyperbolicity, $\pi_1$-diffeomorphism
    Citation: Lerman L. M., Breaking hyperbolicity for smooth symplectic toral diffeomorphisms, Regular and Chaotic Dynamics, 2010, Volume 15, Numbers 2-3, pp. 194-209

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